αs in the Deep-Inelastic Scattering Scheme

Deep inelastic scattering (DIS) data on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{F}_{2}}$$\end{document} structure function accumulated by various collaborations in fixed-target experiments are analyzed in the nonsinglet approximation and within \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {MS} $$\end{document} and DIS schemes. The study of high statistics deep inelastic scattering data provided by BCDMS, SLAC and NMC collaborations, is carried out by applying a combined analysis. The application of the deep inelastic scheme leads to the resummation of contributions that are important in the region of large x values. It is found that using the deep inelastic scheme does not significantly change the strong coupling constant itself but does strongly change the values of the twist-4 corrections.


INTRODUCTION
Currently, the accuracy of data for the deep-inelastic scattering (DIS) structure functions (SFs) allows us to study -dependence of logarithmic corrections based on QCD and power-like (non-perturbative) corrections separately (see, e.g., [1] and references therein).
This article is closely related to those devoted to similar studies, with the main difference being that here we work within the so-called DIS scheme [20], whose application leads to effective resummation of large-x logarithms into the Wilson coefficient functions. We analyze DIS structure function with SLAC, NMC and BCDMS experimental data [21][22][23][24][25][26] at NNLO level of massless perturbative QCD.
As in our previous works [14-17, 27, 28], the function is represented as a sum of the leading twist and the twist-4 terms: 1 (1)

THEORETICAL ASPECTS OF THE ANALYSIS
Here we briefly describe some aspects of the theoretical part of our analysis. For a more detailed description, see [14,27,28]. In the large x-values region gluons do almost not contribute, and the evolution of the twist-2 DIS structure function is determined by the so-called nonsinglet (NS) part.
In this approximation, there is a direct relation between the moments of the DIS structure function and the moments of the NS parton distribution function (PDF) 2 (2) in the form (3) where the strong coupling constant (4) and stands for the Wilson coefficient function. The constant depends on weak and 2 F MS 1 Hereinafter, superscripts pQCD, LT denote the twist two approximation with and without target mass corrections (see, e.g., [27]).
Unlike the standard case, here PDF is multiplied by x.

FIELDS, PARTICLES, AND NUCLEI
electromagnetic charges and is fixed to one sixth for [29].

Strong Coupling Constant
The strong coupling constant is determined from the corresponding renormalization group equation. At the NLO level, the latter is as follows: (5) where (6) At the NNLO level, the strong coupling constant is derived from the equation (7) The expression for I has the form where and are taken from the QCD β-function:

-Dependence of SF Moments
Wilson coefficient function is expressed in terms of the coefficients (hereafter ), which are exactly known (see, e.g., [14]) 3 (8) 3 For the odd n values, coefficients and can be obtained by using the analytic continuation [30][31][32].
The -evolution of the PDF moments can be calculated within the framework of perturbative QCD (see, e.g., [29]): (9) where (10) and (11) are combinations of the NLO and NNLO anomalous dimensions and . For large n (this corresponds to large x values), the coefficients and . So, the terms may lead to potentially large contributions and, therefore, should be resumed.

Factorization
Scale We intend to consider the dependence of the results on the factorization scale caused by (see, e.g., [14]) the truncation of a perturbative series when performing the calculus. The modification is achieved by replacing in Eq. (3) with the expressions in which the scale was accounted in the following way: . Then, Eq. (3) takes the form (12) The function is to be obtained from C by modifying the right-hand side of Eq. (8) as follows: Taking a special form for the coefficient , we can decrease contributions coming from the terms .
3. DIS SCHEME Let us consider the case of the so-called DIS-scheme [20] (it was also called the -scheme [33,34]
The NLO coupling constant obeys the equation

NNLO
In this case we have Eqs. (14) and (15) and additionally (17) that leads to the cancellation of the larger terms in .
However, as seen from our previous efforts we advocate another approach. The analysis is carried out with the moments of SF defined in Eq. (2). Then, for each , SF is recovered using the Jacobi polynomial decomposition method [39][40][41]: (19) where are the Jacobi polynomials, are the parameters to be fit. As usual, the compliance condition is the requirement to minimize the error in restoring the structure functions.
The program MINUIT [42] is used to minimize the variable (20)

RESULTS
We use free data normalizations for various experiments. As a reference set, the most stable hydrogen BCDMS data are used at the value of the initial beam energy GeV. Contrary to previous analyses [14][15][16][17], the cut GeV 2 is used throughout, since for smaller values Eqs. (16) and (18) have no real solutions.
The starting point of -evolution is taken at GeV 2 . This particular value of is close to the average values of covering the corresponding data. Based on previous investigations (see [40]), the maximum number of moments to be accounted for is , and the cut is applied on the data.
We work within the framework of the variable-flavor-number scheme (VFNS) (see [14]). To make it clearer the effect of changing the sign for twist-4 corrections, the fixed-flavor-number scheme with is also used.
As is seen from Table 1 the central values of are fairly the same given total experimental and theoretical errors (see [14][15][16][17]): (21) We plan to study the errors in more detail and present them in an upcoming publication.
From Table 1, it can also be seen that upon resuming at large x values (i.e., in the DIS scheme), the twist-4 corrections become large and negative in this x region. Moreover, it seems that they rise as at large x but this observation needs additional investigations.
The negative values of power-law corrections at large x obtained in DIS scheme leads to the following phenomenon: (part of) power-law terms can be absorbed into the difference between usual strong coupling and QCD analytical one [44] just the same way as it was done at low x values (see [45][46][47][48][49]) in the framework of the so-called double asymptotic scaling approach [50][51][52]. Of course, such a phenomenon was absent in the case of scheme, where using [53] the QCD analytical coupling simply increases the magnitude of twist-4 corrections.

MS
In previous works [2,54], where resuming at large values of x was performed within the framework of the Grunberg approach [55,56], only a decrease in the twist-4 contribution was seen, since the relevant terms were not studied in detail. Therefore, it looks promising if the Grunberg approach will be used in the analysis similar to the present one and thus promote the study in some detail of the twist-4 correction values.

SUMMARY
We have performed fits of experimental data of BCDMS, SLAC and NMC collaborations for DIS SF by resuming large logarithms at large x values into the corresponding coefficient function within the DIS scheme.
It is seen that the resummation does not change the values of the strong coupling constant , though the values of the twist-4 corrections become large and negative contrary to the results obtained in the scheme analyses. We plan to study this phenomenon using another scheme of resuming large logarithms at large x values, such as the Grunberg approach [55,56].